Quater-imaginary base

The quater-imaginary numeral system was first proposed by Donald Knuth in 1955, in a submission to a high-school science talent search. It is a non-standard positional numeral system which uses the imaginary number, 2"i", as its base. By analogy with the quaternary numeral system, it is able to represent every complex number using only the digits 0, 1, 2, and 3, without a sign.

From quater-imaginary to decimal

To convert a digit string from the quater imaginary system to the decimal system, the standard formula for non-standard positional systems can be used. This says that a digit string $d\_3d\_2d\_1d\_0$ in the quater imaginary system can be converted to a decimal number using the formula::$d\_3cdot\; b^3+d\_2cdot\; b^2+d\_1cdot\; b+d\_0$.In this formula b = 2"i" for the quater-imaginary system.

Example

To convert the string $1101\_\{2i\}$ to a decimal number, fill in the formula above::$1cdot\; (2i)^3+1cdot\; (2i)^2+0cdot\; (2i)+1=-8i-4+0+1=-3-8i$

The formula to calculate the decimal number from the quater-imaginary number can be extended when larger strings are used. So to find the decimal counterpart of $1030003\_\{2i\}$ use the formula extended to 7 digits.:$d\_6cdot\; b^6+d\_5cdot\; b^5+d\_4cdot\; b^4+d\_3cdot\; b^3+d\_2cdot\; b^2+d\_1cdot\; b+d\_0$So for $1030003\_\{2i\}$ and with b = 2"i" this gives::$1cdot\; (2i)^6+0+3cdot\; (2i)^4+0+0+0+3=-64+3cdot\; 16+3=-13.$

Powers of 2"i"

When trying to find representations of the numbers from the quater-imaginary system to the decimal system, or vice-versa, the following table is useful:

{| class="wikitable" style="text-align:right"
-!Base 10!!Base 2"i"
-
−1"i"||0.2
-
−2"i"||1030.0
-
−3"i"||1030.2
-
−4"i"||1020.0
-
−5"i"||1020.2
-
−6"i"||1010.0
-
−7"i"||1010.2
-
−8"i"||1000.0
-
−9"i"||1000.2
-
−10"i"||2030.0
-
−11"i"||2030.2
-
−12"i"||2020.0
-
−13"i"||2020.2
-
−14"i"||2010.0
-
−15"i"||2010.2
-
−16"i"||2000.0

Examples

Below are some other examples of conversions from decimal numbers to quater-imaginary numbers.

:$5\; =\; 16\; +\; (3cdot-4)\; +\; 1\; =\; 10301\_\{2i\}$

:$i\; =\; 2i\; +\; 2left(-frac\{1\}\{2\}i\; ight)\; =\; 10.2\_\{2i\}$

:$7\; frac\{3\}\{4\}\; -\; 7\; frac\{1\}\{2\}i\; =\; 1(16)\; +\; 1(-8i)\; +\; 2(-4)\; +\; 1(2i)\; +\; 3left(-frac\{1\}\{2\}i\; ight)\; +\; 1left(-frac\{1\}\{4\}\; ight)\; =\; 11210.31\_\{2i\}$

ee also

* Complex base systems

References

*D. Knuth. "The Art of Computer Programming". Volume 2, 3rd Edition. Addison-Wesley. pp. 205, "Positional Number Systems"